Question

Investment will made to; Company A with probability 0.75 and in Company B with probability 0.25. The annual return from an investment in Company A is approximately normally distributed with mean 15% and standard deviation 4% whereas the annual return from an investment in Company B is approximately normally distributed with mean 10% and standard deviation 2%. Assume that the returns from Companies A and B are independent.

(a) Probability of annual return being between 6% and 18%?

(b Investment earned more than 10% at the end of the year. What is the posterior probability that the investment was in Company B?

(c) %50 of the money is invested to Company A and other half in Company B. Find the expected annual return and the variance of this portfolio.

Answer 1

Solution

Back-up Theory

If X ~ N(µ₁, σ₁²), Y ~ N(µ₂, σ₂²), then

(aX + bY) ~ N(µ, σ²) µ = aµ₁ + bµ₂, and σ² = a²σ₁² + b²σ₂² + 2abCov(x, y) …………………………………………………….. (1)

If X ~ N(µ₁, σ₁²), Y ~ N(µ₂, σ₂²) and X and Yare independent, then

(aX + bY) ~ N(µ, σ²) µ = aµ₁ + bµ₂, and σ² = a²σ₁² + b²σ₂² …………………………………………………….. (1a)

Now to work out the solution,

Let X = Annual return (%) from Company A, Y = Annual return (%) from Company B and

R = Annual return (%). Then, given

R = 0.75X + 0.25Y …………………………………………………………………………….. (2)

X ~ N(15, 4²), Y ~ N(10, 2²), and X and Y are independent …………………………………..(3)

(1a), (2) and (3) => R ~ N(13.75, 3.0414²) ……………………………………………………. (4)

[µ = {(0.75 x 15) + (0.25 x 10)} = 13.75,σ² = {(0.75² x 16) + (0.25² x 4)} = 9.25 = 3.0414² ]

Part (a)

Probability of annual return being between 6% and 18% = P(6 < R < 18)

= P(R < 18) - P(R < 6)

= 0.9189 – 0.0054 [Using Excel Function statistical: NORMDIST(x,mean,standard_dev, cumulative)]

= 0.9035 ANSWER 1

Part (b)

Let C represent the event that investment earned more than 10% at the end of the year and D represent the event that the investment was in Company B.

Then, required probability = P(D/C) ……………………………………………………………………….. (5)

Now, as corollary of Baye’s Theorem,

P(D/C) = {P(C/D) x P(D)}/{P(C)} …………………………………………………………………………….. (6)

P(C/D) = P(Annual return form B is more than 10)

= P(Y > 10)

= 0.5 …………………………………………………………………………………………………………………. (7)

[vide (3), Y is Normally distributed with mean 10 and for a Normal distribution, probability of more than mean is ½.]

P(D) = 0.25 ………………………………………………………………………………………………………… (8)

[Given, Investment will made to; Company B with probability 0.25 ]

P(C) = P(R > 10)

= 0.8912 ………………………………………………………………………………………………………….. (9)

[Using Excel Function statistical: NORMDIST(x,mean,standard_dev, cumulative)]

(6), (7), (8) and (9) => P(D/C) = (0.5 x 0.25)/0.8912 = 0.1403

Thus, the required probability = 0.1403 ANSWER 2

Part (c)

If 50 of the money is invested to Company A and other half in Company B, R = 0.5X + 0.5Y and hence

As shown in (4),

Expected annual return = E(R) = {(0.5 x 15) + (0.5 x 10)} = 12.5 ANSWER 3

Variance of this portfolio V(R) = {(0.5² x 16) + (0.5² x 4)} = 5 ANSWER 4

DONE